Meshless Elastcty Model and Contact Mechancs-based Verfcaton Technque Rfat Aras 1, Yuzhong Shen 1, Mchel Audette 1, Stephane Bordas 2 1 Department of Modelng, Smulaton, and Vsualzaton Engneerng, Old Domnon Unversty, Norfolk, VA, Unted States 2 Insttute of Mechancs and Advanced Materals, Cardff Unversty, Wales, UK Abstract Mesh-based technques are well studed and establshed methods for solvng contnuum bomechancs problems. When the problem at hand nvolves extreme deformatons or artfcal dscontnutes, meshless methods provde several advantages over the mesh-based methods. Ths work dscusses the Movng Least Square approxmaton-based meshless collocaton method for smulatng deformable objects and presents a verfcaton technque that s based on the Hertzan theory of non-adhesve elastc contact. The effectveness of the Hertzan contact theory as a means for verfcaton was frst tested and proven through a well-establshed FEM code, FEBo. The meshless method was mplemented as a reusable component for the SOFA framework, an open source software lbrary for real-tme smulatons. Through expermentaton, the Hertzan theory has been tested aganst SOFA hexahedral FEM and the meshless models wthn the SOFA framework. Convergence studes and L2 error curves are provded for both models. Expermental results demonstrated the effectveness of the mplementaton of the meshless method. 1 Introducton Soft tssue models have a wde range of applcaton areas wth partcular focus on real-tme medcal smulatons. As a result, accurate nteractve modelng of soft tssue s an mportant and well-establshed research feld n contnuum bomechancs studes. A contnuum model typcally reles on an underlyng mesh structure ether n 2D or 3D dependng on the nature and the requrements of the problem. A breadth-frst classfcaton of mesh-based contnuum models ncludes mass-sprng networks [1], Fnte Element Methods [2], Fnte Volume Methods [3], and Fnte Dfference Methods [4]. Orgnally developed for molecular dynamcs problems and astrophyscal smulatons, meshless (mesh-free) methods offer an appealng alternatve to meshbased methods when the problem nvolves large deformatons and mposed ds-
2 contnutes such as cracks and cuts. In ths work, the meshless method s mplemented as a component under the open-source SOFA lbrary, whch focuses on real-tme nteractve smulatons wth an emphass on medcal smulatons. In conjuncton wth the nteractve smulaton requrement, we assume the lnear property of the soft tssue model wthn a specfc range of the stran-stress curve. Ths assumpton s parallel wth the small stran and lnear materal assumptons of the Hertzan contact theory. There have been numerous approaches to modelng mechancs of soft tssues. Among those s the early work of Sederberg and Parry [5] that presented a technque for deformng sold geometrc models n an ntutve free-form manner. The deformatons were based on nterpolatng trvarate Bernsten polynomals, and could be appled ether globally or locally wth volume preservaton. Free-form deformaton s an approxmate and smple method for deformng sold objects; however the lack of physcal bass s grounds for excludng t as an opton for realstc medcal smulatons. The work of Frsken-Gbson [6] modfed the tradtonal voxel based representaton of volumetrc objects and presented a lnked volume representaton that was capable of handlng nteractve object manpulatons such as carvng, cuttng, tearng, and jonng, but stll unable to produce physcally realstc results. An alternatve to volumetrc methods s to use mass-sprng models [7] and membrane based approxmatons that utlze sprng meshes [8]. A sprng mesh s composed of vertces and edges, n whch each edge s realzed as a sprng that connects vertces par-wse, and each vertex s dealzed as a pont mass. Although sprng meshes employ physcal equatons lke Hooke s law, t s dffcult to reproduce specfc elastc materal propertes even wth very careful dstrbuton of sprng stffness through the mesh and suffer from poor numercal stablty. The early work of Bro-Nelsen dscussed a fast adaptaton of fnte element modelng to satsfy speed and robustness requrements n a surgcal smulaton settng [9]. In ths work, the body part was modeled as a 3D lnear elastc sold that conssted of partcles, whch were deformed nto a new shape when forces were appled to the elastc sold. The author ncorporated a technque called condensaton n order to acheve nteractve smulaton. In the fnte element modelng context, condensaton translates nto obtanng a more compact verson of the system model by rearrangng the terms of the matrx equatons. In ths work, the author condensed the equatons by only consderng the fnte element nodes on the surface of the model. A number of recent technques have addressed the fdelty versus effcency trade-off. Another mportant work n the area s the fnte element model based on Total Lagrangan Explct Dynamcs (TLED) by Mller et al. [10]. The dfference between the TLED based fnte element model and other approaches s usng the orgnal reference confguraton of the object to calculate the stress and stran tensors durng a smulaton step. As a result of expressng computatons n the reference coordnates, the authors were able to pre-compute spatal dervatves. The pre-computaton of the spatal dervatves leads to effcency n terms of computatonal resources, whle beng capable of handlng geometrc and materal non-
lneartes. In ther work, the authors employed the central dfference based explct ntegraton rather than the mplct ntegraton scheme. Wth ths choce, they were able to avod solvng the set of non-lnear algebrac equatons that are requred by the mplct ntegraton at each tme step. However, the use of explct ntegraton brngs lmtaton on the tme step sze. The authors justfed ther mplementaton choce by statng that the relatvely lower stffness (Young s modulus) value of the soft tssue relaxes the tme step lmtaton consderably compared to the typcal smulatons nvolvng more stff materal lke steel or concrete. Another attempt to ncrease the computatonal effcency of the elastc model n the context of nteractve smulaton was dscussed n the work of Marchesseau et al. [11]. The authors presented a new dscretzaton method called Multplcatve Jacoban Energy Decomposton (MJED), whch allows the smulaton to assemble the stffness matrx of the system faster than the tradtonal Galerkn FEM formulaton. The authors reported up to fve tmes faster computatons for the St. Venant Krchoff materals. For valdaton purposes, the authors compared the MJED approach to the tradtonal FEM mplementaton n the SOFA framework [12], referred to as the standard FEM mplementaton. FEM technques have domnated the feld of computatonal mechancs n the past several decades. They have been wdely used for modelng physcal phenomena such as elastcty, heat transfer, and electromagnetsm and they heavly rely on the assumpton of a contnuous doman. When the problem doman no longer comples wth ths contnuum assumpton, the ratonale behnd usng an FEM based soluton dsappears. FEM s also not well suted to problems nvolvng extreme mesh dstortons that result n degenerate element shapes, movng dscontnutes that do not algn wth the element edges such as propagatng cracks, and advanced materal transformatons such as meltng of a sold or freezng. To address these ssues, sgnfcant nterest has been developed towards a dfferent class of methods for solvng PDEs, namely meshless or mesh-free methods [13, 14]. The very frst meshless method dated back to 1977 [15] and proposed a smoothed partcle hydrodynamcs (SPH) method that was used to model theoretcal astrophyscal phenomena such as galaxy formaton, star formaton, stellar collsons, and dust clouds. Its meshless Lagrangan nature allowed dverse usage areas besdes astrophyscs such as flud flow, ballstcs, volcanology, and oceanography [16]. Although the SPH method elmnates the necessty of a mesh structure and allows the soluton of unbounded problems, t also has ts lmtatons. Because of ts smplstc kernel based approxmaton scheme, t fals to reproduce even frst order polynomals, resultng n severe consstency problems [13]. To allevate ths problem, methods that utlze movng least squares (MLS) approxmatons have been developed. The frst work that used MLS approxmatons n a Galerkn method s the work of Nayroles et al. [17], whch was refned by Belytschko et al. [18] and named Element-Free Galerkn (EFG) method. Ths class of methods, dfferent from the SPH method, use shape functons n approxmatons that are essentally corrected versons of compact supported weght functons. The shape functons 3
4 are obtaned by frst representng the approxmaton as a product of a polynomal bass and a vector of unknown coeffcents. Then, a functonal s created by takng the weghted sum of square of the approxmaton error. By takng the dervatve of ths functonal wth respect to the unknown coeffcents and settng t to zero for mnmzng the approxmaton error, we obtan a set of equatons that are reorganzed to solve for the MLS shape functons. The order of consstency of the MLS approxmaton scheme depends on the order and completeness of the used bass functon. If the bass functon used n the approxmaton s a complete polynomal of order k, then the MLS approxmaton s sad to be k th order consstent. Ths fact makes the MLS based approxmatons more consstent than the SPH method. Another technque that has used the MLS approxmaton s the work of Mueller et al. [19] and forms the bass of the meshless method dscussed n ths paper. In ths work, the authors calculated the spatal dervatves of the deformaton gradent only at the partcle locatons smlar to the meshless collocaton methods. Ths technque s capable of smulatng a wde range of materal propertes from very stff materals to soft ones and also handles plastc deformatons as well. Horton et al. [20] proposed a new knd of meshless method named meshless total Lagrangan explct dynamcs method. In ths work, the authors extended ther prevous TLED algorthm [10] to the meshless dscretzaton methodology by precomputng the stran-dsplacement matrces. Ther method s a fully explct method, meanng not requrng mplct tme ntegraton and costly soluton of large system of equatons. Dfferent than our nodal ntegraton approach, ther proposed algorthm ntegrates over a regular background grd wth sngle ntegraton pont per cell. In ths work, we present the detals of the MLS approxmaton-based meshless collocaton method for soft-tssue deformaton. Specfc contrbutons are SOFA mplementaton of the presented meshless method, a new verfcaton technque wth the Hertzan non-adhesve contact theory, and verfcaton of the algorthm wth well-establshed FEM code and our SOFA component that mplements the presented meshless method. The rest of the paper s organzed as follows. Secton 2 descrbes the detals of the meshless elastcty algorthm, ncludng node dstrbuton, constructon of support domans, used weght functon, and calculaton of nodal smulaton values such as mass, volume, and densty. Operatons nvolved n the MLS approxmaton of the deformaton gradent are dscussed, followed by the nternal elastc force calculaton procedure. Secton 3 descrbes the mplementaton detals of the presented meshless method n SOFA and presents the verfcaton technques for the elastcty model along wth the vrtual experment setup and the results. Secton 4 concludes the paper and dscusses possble future extensons to the presented work.
5 2 Algorthm 2.1 Dscretzaton of the Contnuum Node dstrbuton s the frst step n the presented meshless collocaton algorthm, whch supports both regular and herarchcal dstrbuton of the nodes through the smulaton doman. In the case of a smulaton doman wth a regular geometrc shape, regular dstrbuton of the nodes s the natural choce. On the other hand, f the smulaton doman has a complex geometry, the regular dstrbuton smply becomes napplcable. In ths case, we sample the volumetrc smulaton doman bounded by the complex boundary surface by herarchcally samplng the volume. In the work of Pauly et al.[21], the authors used a balanced octree data structure to dstrbute the nodes nsde the volume. In ths work, we frst tetrahedralze the smulaton doman wth well-establshed computatonal geometry lbrares lke TetGen [22] and CGAL [23] and then use the set of vertces of the tetrahedra as the meshless node locatons. In ths way, smlar to graded fnte element meshng technques, we can have hgher node densty close to the doman boundary and fewer nodes towards the nteror of the volume where the materal s contnuous. Meshless methods represent a deformable body by a cloud of partcles wth overlappng support domans. Quanttes such as mass, volume, support sze, stran, and stress are stored and updated per partcle for the duraton of the smulaton. In ths work, the support domans of the partcles are sphercal and ther rad are computed by fndng the average dstance of the central node to ts k-nearest neghbors. For effcent neghborhood search purposes, a k-d tree data structure s used. Weght (kernel) functon n the meshless method context s an element that descrbes the way meshless nodes affect each other and how the materal values of the contnuum such as mass, volume, and densty are dstrbuted among the nodes. The neghborng partcles that fall nsde the support doman of a central partcle are weghted usng the polynomal kernel functon wth r j 2 4 6 1 3rj 3 rj rj, rj 1 wr ( j) 0, rj 1 j x x, where x and x j are the current locatons of the neghbor- h ng and central partcles, respectvely and h s the support radus of the central partcle. The mass and densty of a meshless node are assgned at the begnnng and kept fxed throughout the smulaton. The mass values are ntalzed wth (1)
6 m sr, (2) where ρ s the materal densty value, r s the average dstance of the th node to ts k-nearest neghbors, and s s a scalng factor that s chosen so that the average of the assgned denstes s close to the actual materal densty. The assgned mass value of a meshless node s spread around the node wth the kernel functon. Therefore the densty of a meshless node s calculated after the mass allocaton step by takng the weghted average of the masses of the neghborng nodes m w( r ) (3) 3 j j j In ths work, spatal ntegraton s performed through the nodal ntegraton technque. Compared to other spatal ntegraton technques that utlze a background mesh or grd wth multple ntegraton ponts per regon, nodal ntegraton s fast and effcent wth the added dsadvantage of decreased stablty. We calculate the spatal dervatves of the deformaton gradent only at the partcle locatons smlar to the meshless collocaton methods. 2.2 Movng Least Square (MLS) Approxmaton wth Taylor Seres In contnuum sold mechancs, the elastc stresses nsde a deformable body are computed based on the dsplacement vector feld, whch s typcally defned by a deformable model wth two confguratons named the reference and the current confguratons, respectvely. The coordnates of the partcles n the reference and current confguratons are represented by the materal X ( X, Y, Z) and spatal x ( xy,,z) coordnates respectvely. The dsplacement vector of a partcle s u u u u x X. therefore defned as T X Y Z The Green-St.Venant stran tensor s used to measure the stran T T u u u u, (4) where the gradent of the contnuous dsplacement vector feld u s essentally the dervatves of ( ux, uy, u z ) wth respect to ( X, Y, Z ) arranged n the Jacoban format. For an sotropc lnear-elastc materal, the stran s mapped to the stress by the C tensor that approxmates the materal propertes and s composed of two ndependent coeffcents, Young s Modulus and Posson s Rato C. (5) We need the partal dervatves of the dsplacement vector feld n order to compute the stran, stress, and the nternal elastc forces appled to the meshless partcles. Movng Least Squares (MLS) approxmaton s used to compute ths gradent ( u ).
7 For a central meshless partcle and ts neghbor j, the value of u at the locaton of j can be approxmated by the frst order Taylor expanson as u ux u.( X X ) X x j x j x. (6) The weghted sum of squared dfferences between the dsplacement vector and ts approxmaton obtaned from the equaton (6) gves the error measure of the MLS approxmaton 2 e ( ux u ) ( ) j x w r j j. (7) j The expanded equaton of the error measure for a partcle s therefore obtaned by ux ux ux 2 e ( ux ( )X ( ) ( ) ) ( ) j Yj Zj ux w r j j. (8) X Y Z j We want to mnmze ths error measure for some values of ( u x )/ X, ( u x )/ Y, and ( u x )/ Z, therefore we set the dervatve of the error measure e wth respect to the partal dervatves of the dsplacement vector to zero, resultng n three equatons for three unknowns T ux ( X j X)( X j X) w rj ux u ( ) j x j w rj j X X. (9) X j The coeffcent of the partal dervatve on the left-hand sde of the equaton (9) s the 3x3 matrx called the moment matrx (A). A can be nverted and premultpled wth both of the sdes of the equaton for computng the partal dervatves. 2.3 Force Calculaton The elastc body forces that are appled to the ndvdual partcles n the meshless collocaton method are calculated through the stran energy densty, whch s a functon of the partcle dsplacements. For a partcle wth volume v, stran, and stress, the stran energy densty becomes 1 U v ( ). (10) 2 The elastc force per unt volume at a meshless node s locaton s the negatve drectonal dervatve of the above stran energy densty wth respect to ths node s dsplacement. The forces appled to the partcle and ts neghbors j are then
8 f U v u u x f U v xj x j u u xj. (11) Eq. (11) s terated over all partcles and the total force appled to a partcle s the sum of all the forces wth the same ndex as that partcle. These force components are obtaned by usng the Green-Sant-Venant stran tensor, whch measures the lnear and shear elongaton. In case of a volume nvertng dsplacement though, ths stran becomes zero. In order to ntroduce restorng body forces n cases of volume nverson, Muller et al. [19] added another energy term to the system that penalzes devatons from a volume conservng transformaton 1 2 U kv( J 1). (12) 2 In ths energy term, J s the Jacoban of the dsplacement vector feld mappng and v k s the volume restoraton constant. Although n our experments we have not worked on cases that cause the nverson of the volume, the effect of ths term and the volume restoraton constant parameter n cases of large deformatons s yet to be examned. For each of the meshless nodes, these force components are accumulated n the force vectors and then passed to the SOFA tme ntegraton module. The modular mplementaton of the SOFA lbrary allows the use of dfferent tme ntegraton schemes wthout changng the actual mplementaton of the algorthm. In ths work, we used the 4 th order Runge-Kutta ntegraton scheme. 3 Implementaton and Verfcaton Technques In modelng and smulaton studes, verfcaton and valdaton of the model s a crucal step. For nteractve smulatons wth a focus on tranng n partcular, t s mportant to valdate the behavor of the deformable body n order to prevent false learnng outcomes. Specfcally, realstc contact representaton and force feedback are sgnfcant features for ensurng the valdty of physcally based smulators. 3.1 Meshless Method Implementaton n SOFA SOFA [12] s an open-source object-orented software lbrary that s targeted towards nteractve medcal smulatons. SOFA has a modular structure that allows users to quckly prototype smulaton scenes wth ready-to-use components. Its object-orented, modular archtecture makes t easy for developers to extend the
9 functonaltes of the lbrary by dervng new components from the exstng ones. The meshless elastc model descrbed n ths work has been mplemented as a force feld component, whch s easly nterchangeable wth the exstng force feld components such as hexahedral fnte elements or mass-sprng networks. The requred component nterfaces such as ntalzaton and force accumulaton were mplemented accordng to the algorthm steps descrbed n the prevous secton. Due to the nature of the presented verfcaton method, accurate contact handlng s essental to obtan precse results. Unfortunately, because of the approxmatng nature of the meshless methods, mposng Drchlet boundary condtons s a challengng task on ts own [24]. In ths work, we have followed the approaches that were orgnally adopted n the SOFA lbrary. In SOFA, constrants are flter lke components, whch cancel out the forces and dsplacements appled to ther assocated partcles. For example for fxed node boundary condtons, SOFA's fxed constrant component s used to attach a meshless partcle to a fxed pont n the current confguraton. SOFA has support for several contact handlng methods such as penalty-based and constrant-based methods. Among these, the constrant-based methods are more appealng than the former class of methods because they use Lagrange multplers to handle complex constrants and produce physcally accurate results wth the addtonal computaton cost. Lagrange multplers wth unlateral nteracton laws are used to handle complex constrants. The constrants depend on the relatve postons of the nteractng objects, whch are the meshless partcles and the sphercal rgd ndenter n our case [12]. 3.2 Contact Mechancs Theory Hertzan theory of non-adhesve elastc contact [25] defnes analytcal solutons for the nteracton of elastc half-spaces wth smple shapes n terms of appled force and object ndentaton. For example, the amount of ndentaton of an elastc half-space under a sphercal load s gven by 4 * 3 2 F E Rd, (13) 3 where F s the vertcal force appled on the sphercal load, R s the radus of the * sphercal load, d s the ndentaton amount, and E s the combned Young s E, E ) modulus of the two materals and calculated usng the Young s modul ( 1 2 and Posson s ratos ( v 1, v 2 ) of the two materals as 1 1 v 1 v. (14) 2 2 1 2 * E E1 E2
10 The Hertzan theory assumes 1) small strans wthn the elastc materal, 2) much smaller area of contact compared to the areas of the objects n contact, and 3) contnuous and frctonless contact surfaces. The Hertzan theory s an mportant steppng stone n the feld of contact mechancs; therefore there have been numerous fnte element analyss studes about the subject that use both research and commercal fnte element code [26-29]. 3.3 Verfcaton Experments In order to verfy the usablty of the Hertzan contact theory as a means of verfcaton of soft-tssue deformaton, we frst conducted experments usng wellestablshed fnte element code. FEBo s an open-source software sute that s prmarly targeted towards bomechancs and bophyscs problems wth a specfc focus on nonlnear large deformaton problems n bosold mechancs [30]. FEBo provdes several models and optons to represent the non-adhesve Hertzan contact theory. In our experments, we selected the facet-to-facet sldng algorthm that s based on Laursen s contact formulaton [31]. In ths algorthm, the contact constrants are enforced through Lagrange multplers. The FEBo experment was setup by defnng the fxed-poston boundary condtons of the deformable block at ts bottom and sde faces, facet-to-facet sldng contact between the top face of the deformable block and the rgd sphercal ndenter, and the sphere s ndentaton amount. The smulaton was run for 10 tme steps of 0.1s each and the smulaton runtme took over 4 mnutes. The node at the mddle of the top of the deformable block was tracked for the vertcal dsplacement and the vertcal component of the contact force. We compared the obtaned load-dsplacement curve to the theoretcal soluton, whch were n very good agreement (Fgure 1), therefore verfyng the usefulness of the Hertzan contact theory as a verfcaton method. Fgure 1 Comparson of the FEBo FEM Code and the theoretcal soluton of the Hertzan non-adhesve frctonless contact theory.
11 The contact mechancs experment was setup as a SOFA scene. In order to assess the ground-truth performance of the contact handlng n SOFA, the rectangular deformable block was represented wth the hexahedral fnte element model n addton to the descrbed meshless method. The valdaton of the hexahedral FEM mplementaton of SOFA s studed by Marchal et al. [32]. For a gven sphere radus, smulatons were performed for varyng force values (F) appled to the sphercal load (Fgure 2). Wth the appled force, the rgd sphere comes nto contact wth the block and deforms t. The vertcal velocty of the sphere s montored and the ndentaton of the materal (d) s measured when the sphere comes to rest. Ths F-d par s compared to the theoretcal soluton. (a) Fgure 2 (a) Intal setup of the ndentaton experment for the SOFA FEM model, (b) the close-up vew of the ndented deformable materal. SOFA allows the user to track and montor smulaton values of ndexed partcles. The meshless nodes are dstrbuted unformly nsde a cubcal volume wth 2m long edge length. The ndentaton experment s repeated for several dstrbuton confguratons, whch play crtcal role especally for the MLS approxmatonbased collocaton methods. The convergence rate n the L2 (vector) error norm of the force-ndentaton pars wth respect to the theoretcal values (Fgure 3) s nvestgated. The effect of dfferent dstrbuton schemes on the accuracy, stablty, and performance of the meshless collocaton methods s yet to be examned. In our mplementaton, the number of neghborng nodes s lmted to 16 for each of the meshless nodes. (b)
12 Fgure 3 Error n the L2 norm wth respect to the theoretcal soluton as functons of total number of the degrees of freedom for the (a) FEM model and (b) meshless method. We compared the SOFA FEM mplementaton and the meshless collocaton method wth close accuracy (Fgure 4). For our meshless collocaton method wth nodal ntegraton, we used an explct tme ntegraton scheme wth a tme step of 0.001s wthout any stablty problem. For the SOFA FEM mplementaton, we had to use mplct ntegraton wth a tme step of 0.01s. The calculatons were performed wthn the SOFA applcaton on a sngle Intel Core 5 CPU runnng at 2.67 GHz wth 16 GB of RAM under Wndows 7 operatng system. The SOFA FEM mplementaton took 195ms of calculatons per tme step, whereas the meshless method consumed 20.11ms for calculatons per tme step. Therefore, the meshless collocaton mplementaton n SOFA (along wth other SOFA related operatons such as collson detecton) s roughly 25 tmes slower than the real-tme operaton, whch s slghtly better than the 30 tmes slower performance reported by the Meshless TLED algorthm [20]. The calculaton speed of the meshless collocaton algorthm s governed by the number of partcles and the number of neghbors assgned to each partcle. Fgure 4 Comparson of the SOFA FEM mplementaton and the meshless collocaton method wth close ndentaton accuracy and the theoretcal soluton.
13 We also performed a mesh convergence study for the meshless collocaton method by nvestgatng the convergence of the ndentaton amount to the theoretcal value for a fxed amount of force (Fgure 5). After around 6000 partcles, the ndentaton value converges to the theoretcal ndentaton value. Fgure 5 Convergence of the ndentaton value wth ncreasng number of meshless partcles. 4 Conclusons and Future Work The dscussed Movng Least Square approxmaton-based meshless method may be useful n contnuum problems wth extreme deformatons or movng dscontnutes such as cracks or cuts. The presented algorthm for the meshless collocaton method was mplemented as a component for the open source SOFA lbrary, whch s prmarly targeted at real-tme medcal smulaton. Contact mechancs-based verfcaton experments were conducted wth FEBo FEM code, SOFA hexahedral FEM method, and the presented meshless collocaton method. The convergence study results and L2 error norm curves are promsng for the meshless deformable model. For the meshless collocaton methods, the nfluence of the meshless node dstrbuton and the sze of the nodal support rad of the nodes are not well studed. The fndngs of our own experments also suggest further studes of these mportant aspects of the meshless collocaton methods.
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